The frequency \(\omega _0\)  could be estimated with Eq. (REF ) in the \(M\mu \ll 1\)  limit. Then the right-hand side is a function of \(M\mu \)  and the angular momentum \(a_*\) . With this approximation, we plot the GW emission luminosity as a function of \(M\mu \)  in Fig. REF . The angular momentum \(a_*\)  for each curve is chosen to be the corresponding \(a_{*C}^{nlm}\)  given in Eq. (REF ). In the figure, numerical results with \(\widetilde{l}\le 5\)  are compared to the asymptotic expressions given in Eqs. (REF ). They agree at \(M\mu \ll 1\) , showing the consistency of our calculation. At \(M\mu =0.1\) , the asymptotic expression for the \(\lbrace 0,1,1\rbrace \)  mode is 3 times larger than the numerical result. Away from the region \(M\mu \ll 1\) , the asymptotic expression could overestimate the emission power by a factor as much as 40 for \(l=m=1\)  modes and \(2.5\)  for \(l=m=2\)  modes. Also plotted in the figure is the numerical solution from Ref. [1]}, which keeps only the first two terms in the \(\widetilde{l}\)  summation for the \(\lbrace 0,1,1\rbrace \)  mode. It differs from our result by at most \(39.0\%\) , exhibiting the good convergence of the \(\widetilde{l}\)  summation.
